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I don't know the answer to this question. I was having fun with my General Topology book (LIMA, Elon. Elementos de Topologia Geral) and began wondering.

This question is intended to be the most general possible.

With "a isoperimetric problem" I mean a 4-tuple $(\Omega,\tau,f,K)$, where f is a measure function for the topological space and $K$ is a group with an operation $+$ whose elements I'll call scalars. There is an isoperimetric question in this 4-tuple that can be answered, and this question shows me the maximum measure of frontier of an open set when mantaining constant its interior measure.

I believe creating curves may make me get out of topology and definitely enter geometry. But I don't want that! What I want is to make an isoperic problem inside topology.

Can I make curves without depending on a metric?

What could be this isoperimetric question?

Thanks.

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    Could you please explain what $\Omega$, $\tau$, $f$, and $K$ would be for the standard isoperimetric problem of a curve of fixed length in the Euclidean plane? And in the proposed generalization, is $(\Omega, \tau)$ a topological space (with underlying set $\Omega$ and $\tau$ the topology)...? – 2017-01-03
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    thanks for the reply! $(\Omega, \tau)$ is a topological space. I infer exactly the wikipedia definition of isop. problem. https://en.wikipedia.org/wiki/Isoperimetric_inequality. maybe my problem is how to create curves. how can I create curves? – 2017-01-03
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    @AndrewD.Hwang, sorry, I guess I misread. The answer for your question is (Euclidean space,regions,euclidean metric,positive real numbers) – 2017-01-04
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    I really would like to be more clear about this, but maybe topology is still too sofisticated to me :( – 2017-01-04
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    Nothing brief comes to mind; one idea would be to fix a measure on $\tau$, then define the measure of the boundary of an open set by taking the $\liminf$ as $\epsilon \searrow 0$ of $1/\epsilon$ times the total measure of coverings of the boundary by open sets of diameter at most $\epsilon$. Offhand I don't know what kinds of technical conditions should be imposed on the measure and topology to make things "work more or less as expected". – 2017-01-04
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    I liked that! Measure theory is really new to me. When you said measure you are saying "Lebesgue Measure" or the general word fot measures which may include "Lebestue Measure"? – 2017-01-04
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    There are axiomatic versions of "measure", but practically speaking, I expect you'd want something not far from Lebesgue measure, defined on a vector space or some space locally modeled by a vector space (such as a connected Riemannian manifold). You might also want the space to be homogeneous (or something even more stringent) under the automorphism group of whatever structure is imposed. The Riemannian setting has been studied for complete, simply-connected manifolds of constant curvature. Do you know about [geometric measure theory](https://en.wikipedia.org/wiki/Geometric_measure_theory)? – 2017-01-04
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    No, I didn't know this theory existed! And it is beautiful! Thank you so much. What books about this theory would you recommend? Do you think there are books in portuguese about it? – 2017-01-04
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    I believe Frank Morgan's book _Geometric Measure Theory_ is undergraduate-level, though I have no idea whether it's been translated into Portuguese.... – 2017-01-04

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